research partners    
Ph.D. students    
consortium members    

Piotr Gwiazda
Institute of Applied Mathematics and Mechanics
University of Warsaw
Banacha 2,
02-097 Warsaw
phone: +48 22 5544551
fax: +48 22 5544700

The issue of regularity for systems of complex flows based on the Navier-Stokes equations
Wojciech Zajączkowski (IMPAN) and Gregory A. Seregin (Mathematical Institute University of Oxford)
The project is carried out by Magdalena Bogdańska at IM PAN
This project is devoted to the problem of regularity of weak solutions to the Navier-Stokes equations. We can distinguish the following directions:

(1) formulation of Serrin type conditions for some special solutions
(2) problem of existence of global regular solutions to the Navier-Stokes equations which remain close to either the two-dimensional solutions, or to the axially symmetric solutions, or to the helicoidal solutions.
(3) development of new techniques as Besov spaces, BMO spaces with an essential use of the harmonic analysis
(4) examining of the existence of global regular solutions of the Navier-Stokes equations coupled with the heat equation, equations describing internal properties of the fluid, and also the same for the nonhomogeneous Navier-Stokes equations and magnetohydrodynamics.

Examining the existence of regular solutions to the Navier-Stokes equations which are close to the axially symmetric solutions we need the technique of weighted Sobolev spaces. Therefore we need the solvability of boundary and initial-boundary value problems for the Laplace equation, the heat equations and the Stokes system in the weighted Sobolev spaces.

[1] W. Zajączkowski (2008) Zapiski Nauchn.Sem POMI.
[2] J. Y. Chemin, I. Gallagher, M.Paicu, (2008) arXiv:~0807.1265v1
[4] G. Seregin, (2006) arXiv:0607537v2